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YOU are better than YOU think. Show
yourself how:
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
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Appendix D
What to do in School and Why
Previous: C How to Read
A Why Go or Attend
Education is for both sons and daughters. The assumption that someone else
will provide home and shelter for us is too often false. So you should get the
most out of your schooling and education. Your ability to get a job, your future
earnings and your future joys and companions may all depend on what you learn in
and after school. All this may also depend on factors beyond your control (wars,
social conditions, the economy or parents). The suggestions provided below are
for some, not all eventualities.
School can be a place where you learn to master rule and pattern-based
thought (logic) and where you master further skills and knowledge. You are in
school to learn about the wide range of human knowledge and behavior. Try to
understand whatever you might be asked to study or do. Think for yourself and
ask why you or others do this or that.
In school, you should look for the ideas new to you and for the ideas worth
repeating to others. That is what makes a class or a subject worthwhile. Again,
ideas which you have already seen are reassuring and comforting - not much work
is needed here, but you only learn from the ideas new to you. Look and search
for them.
What you see in school and in books represents the experience of others. When
you are observant, you can learn from the experience, skills and mistakes of
others instead of your own.
History courses, English (or literature) courses, science courses and some
realistic novels offer this experience secondhand. Secondhand knowledge of
hardship is preferable to first hand. Ask your teachers, relatives and the
others about the joys and the difficulties they met or foresee. Learn from their
experience. Ask for opinions. Guidance from others requires the statement of
opinions. Get two or more opinions even if you liked the first. Seek and
politely allow opinions different from your own. Different points of view may
sharpen or change yours.
B Health and Social Skills
You are also in school to meet people and to learn how to mix or socialize
with others. To this end, join a club or group activity. See how people,
including yourself behave in groups. After school, the opportunities to mix may
decline - or you may not develop the habit of mixing and socializing. Choose
activities you like. Try one, two or several, but leave enough time for your
studies and for special events. Suggestion: each week, get three or more hours
of physical exercise. This exercise could come from physical labor. Or, you may
find a sport or activity which you can do now and later. This exercise should
build your health without risking it and without damaging it.4
4I
saw in a university soccer match or practice in 1984, a player with a small
cast on his leg. I thought he was risking permanent damage.
In 1990, after twenty years of cross-country skiing for exercise etc, I began
to ski in colder and colder conditions without fear. The eventual result was a
deep frostbite to my cheeks, an area difficult to protect. Then for five
years, exposure to the cold was an unpleasant experience followed by hours of
pain or discomfort - a distraction from work and play that is not recommended.
C Suggestions for Learning
By law you are required to attend school. Make sure your time is not wasted.
Make sure that some of your courses are with helpful, hard-driving, teachers.
Ask them for advice on what to do or what could be useful to you. Further advice
follows. It repeats in part advice given in previous appendices.
- Look for the ideas new to you and for ideas worth repeating. When you are
preparing for a test or for a future lesson, your studying is done when you
can find no ideas new to you.
- Try to remember the names of places, people and ideas. You can use the
names in conversations, essays and tests later.
- Learn to read precisely what is written. This skill will serve you well.
It gives you more independence both in class and when you leave school. It
may allow you to learn at your own pace.
- Learn to take notes. When no textbook is present, note-taking skills will
be needed. When a textbook is present, look at it first. (Reading it in
advance may allow you to take fewer notes and understand lessons better.)
- Learn to type. Today, computers are used in all areas of office work
bureaucracy and technology. These computers are controlled by keyboards.
Accurate, if not fast, typing skills will make your exposure to computers
and report writing more pleasant.5
5This
advice is valid now. Advances in computer technology - the introduction of
voice-controlled dictation/computer systems - may make part of this advice
stale or obsolete.
- Get careful thinking skills. That is, master the use of rules and
patterns. Every area of skill and knowledge offers rules and patterns which
you might follow. Learn to read exactly what each says. To follow, to agree
or to disagree with rules, you need to understand exactly what they say and
exactly what they don't say.
- In high school take courses which provide immediate job skills such as
auto-mechanics, typing, metalwork, woodwork, drafting etc. Master arithmetic
and learn to read and write carefully. Employers want skilled workers. They
are easier to train and worth keeping. Even if you are planning a college
education, practical job skills could get you a summer job. They may allow
you to work and pay for part of a college education. Care is required to
take the best and avoid the worst of the academic and non-academic courses
in your school.
- Take English or master another language of your choice, well. This
includes reading, writing, speaking and reasoning. When you write, tell a
story, describe what is, or present an opinion or defend one. Watching for
ideas worth repeating, will help.
- Take history courses. Courses with ideas new to you are worth taking. If
possible, avoid history courses which only promote the group, state or
country in which you live. History courses tell us about the experiences and
mistakes of past, if not present generations.
- Read newspapers which do not (always) glorify the nation or group to which
you belong. Contrary opinions make us think. So look for and read newspapers
with views you occasionally find disagreeable.
- A little uncertainty in the words of a teacher leaves room for thought and
the practice of thinking skills.
Appendices with (repetitive) advice for Students: [ B How to Learn ] [ C. How to Read ] [ D. What to do in School ] [ PS. Study Tips ] [ PS: Time and Effort ] [ E. How to Study Math and Why ]
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www.whyslopes.com
Volume 2, Three Skills for Algebra -
Preview, starter & further lessons for logic and algebra
to (i) improve work & study skills; (ii) to to ease or avoid
algebra (math) fears & difficulties; and (iii) to fill gaps in the
exposition of mathematics.
Foreword, Chapters and Appendices follow.
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6 Old Language
7 Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I. Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning
Words Before Symbols:
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
Complex number: starter lesson
Solving Linear Equations:
A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.
(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v) (½)x + 8 = 24½
(vI) (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With
Parameters
Problem Solving with Linear
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
C. Solving Linear Eq'ns
without
Stick Diagrams
D.
Problems in
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems
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